 Author: STEFAN NICOLA Language: English, Spanish, German Country: Eritrea Genre: Biography Pages: 692 Published (Last): 15.07.2016 ISBN: 550-3-80917-897-5 ePub File Size: 27.36 MB PDF File Size: 18.14 MB Distribution: Free* [*Regsitration Required] Downloads: 40794 Uploaded by: CHAU Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our User Agreement and Privacy Policy. See our Privacy Policy and User Agreement for details. Published on Mar 23, SlideShare Explore Search You.

How is the third term related to the coefficient of the middle term? Explain how you arrived at your answer. What mathematics concepts or principles did you apply to come up with your answer? How do you describe a perfect square trinomial? Did you get the same answer? Make It Perfect!!! Determine a number that must be added to make each of the following a perfect square trinomial. Is there an easy way of expressing a perfect square trinomial as a square of a binomial?

If there is any. Such a mathematical sentence will be used to satisfy the conditions of the given situation. Observe the terms of each trinomial. How did you express each perfect square trinomial as the square of a binomial? Let us further strengthen your knowledge and skills in mathematics particularly in writing perfect square trinomials by doing the next activity.

Were you able to figure out how it can be easily done? Compare your answer with those of your classmates. Another method of solving quadratic equation is by completing the square. Use the diagram to answer the following questions.

Divide both sides of the equation by a then simplify. Using the equation formulated. The area of the cemented part is m2. How would you represent the length of the side of the car park? How about the width of the cemented portion?

Finish the Contract! The shaded region of the diagram at the right shows the portion of a square-shaped car park that is already cemented. Write the equation such that the terms with variables are on the left side of the equation and the constant term is on the right side. What equation would represent the area of the cemented part of the car park? Can you tell why the value of k should be positive?

Are you ready to learn about solving quadratic equations by completing the square? Check the solutions obtained against the original equation. Solve the resulting linear equations. Add the square of one-half of the coefficient of x on both sides of the resulting equation. Solve the resulting quadratic equation by extracting the square root.

The left side of the equation becomes a perfect square trinomial. Divide both sides of the equation by 2 then simplify. Express the perfect square trinomial on the left side of the equation as a square of a binomial. Add 18 to both sides of the equation then simplify.

Add 41 to both sides of the equation then simplify. Notice that 25 is a perfect square. Add to both sides of the equation the square of one-half of Your goal in this section is to apply the key concepts of solving quadratic equations by completing the square. Complete Me! Find the solutions of each of the following quadratic equations by completing the square. Was it easy for you to find the solutions of quadratic equations by completing the square?

Then find the solutions to the equation by completing the square. Were you able to come up with the right representations of the area of the shaded region of each figure?

Were you able to solve the equations formulated and obtain the appropriate measure that would describe each figure? Represent then Solve!

Using each figure. Do all solutions to each equation represent a particular measure of each figure? What expressions represent the length. Do you agree that any quadratic equation can be solved by completing the square?

If you are to choose between completing the square and factoring in finding the solutions of each of the following equations. The first few parts of his solution are shown below. Explain and solve the equation using your preferred method. How would you represent the dimensions of the cardboard? You are going to think deeper and test further your understanding of solving quadratic equations by completing the square.

Can she use it in finding the solutions of the equation? Explain why or why not. An open box is to be formed out of a rectangular piece of cardboard whose length is 8 cm longer than its width.

What are the dimensions of the rectangular piece of cardboard? You will be given a practical task which will demonstrate your understanding of solving quadratic equations by completing the square. In what year did the average weekly income of an employee become Php Using the mathematical sentence formulated. Write a quadratic equation that would represent the volume of each box.

What is the length of the box? How about its width and height? Design Packaging Boxes!!! Perform the following. If the box is to hold cm3. Designing Open Boxes 1. Make sketch plans of 5 rectangular open boxes such that: What new insights do you have about solving quadratic equations by completing the square? Designing Covers of the Open Boxes 1. This lesson was about solving quadratic equations by completing the square. Make sketch plans of covers of the open boxes in Part A such that: Solve each quadratic equation by completing the square to determine the dimensions of the materials to be used in constructing each box.

The lesson provided you with opportunities to describe quadratic equations and solve these by completing the square.

Start Lesson 2D of this module by assessing your knowledge of the different mathematics concepts previously studied and your skills in performing mathematical operations. Work with a partner in simplifying each of the following expressions. These knowledge and skills will help you in understanding solving quadratic equations by using the quadratic formula. If you did. Did you arrive at the same answer? Were you able to simplify the expressions?

How would you describe the expressions given? Follow the Standards! Write the following quadratic equations in standard form. Then identify the values of a. Are there different ways of writing a quadratic equation in standard form? Which expression did you find difficult to simplify? How do you describe a quadratic equation that is written in standard form? Were you able to write each quadratic equation in standard form?

Were you able to determine the values of a. You need this skill for you to solve quadratic equations by using the quadratic formula. How did you simplify each expression? How would you represent the dimensions of each garden? How would you describe the equation formulated in item 4? How are you going to find the solutions of this equation? Bonifacio would like to enclose his two adjacent rectangular gardens with What mathematical sentence would represent the length of fencing material to be used in enclosing the two gardens?

How about the mathematical sentence that would represent the total area of the two gardens? The gardens are of the same size and their total area is m2. Do you think the methods of solving quadratic equations that you already learned can be used to solve the equation formulated in item 4?

Did the activity you just performed capture your interest? Were you able to formulate mathematical sentence that will lead you in finding the measures of the unknown quantities? What equation will you use in finding the dimensions of each garden? Why do the Gardens Have to be Adjacent? How will you find the dimensions of each garden?

What are the solutions of the given equation? Are you ready to learn about solving quadratic equations by using the quadratic formula? Compare your work with those of other groups.

Did you obtain the same solutions? Do you think the equation or formula that would give the value of x can be used in solving other quadratic equations? Using the resulting equation in item 5. How would you describe the solutions obtained? What equation or formula would give the value of x?

Lead Me to the Formula! Work in groups of 4 in finding the solutions of the following quadratic equation by completing the square. How did you use completing the square in solving the given equation? Show the complete details of your work. This formula can be derived 2a by applying the method of completing the square as shown on the next page. Write the equation in standard form. Simplify the result if possible then check the solutions obtained 2a against the original equation.

Is the Formula Effective? Find the solutions of each of the following quadratic equations using the quadratic formula. Your goal in this section is to apply the key concepts of solving quadratic equations by using the quadratic formula.

Which of the solutions or roots obtained represents the width of each plywood? Bonifacio cuts different sizes of rectangular plywood to be used in the furniture that he makes. What quadratic equation represents the area of each piece of plywood? Write the equation in terms of the width of the plywood. Then determine the values of a. What is the length of each piece of plywood?

Some of these rectangular plywood are described below. Is there any equation whose solutions are equal? Solve each quadratic equation using the quadratic formula. Plywood 2: Was it easy for you to find the solutions of quadratic equations by using the quadratic formula? Were you able to simplify the solutions obtained?

Cut to Fit! Read and understand the situation below then answer the questions that follow. Write each quadratic equation formulated in item 1 in standard form.

Plywood 3: The perimeter of the plywood is 10 ft. Plywood 1: The length of the plywood is twice its width and the area is 4. Is there any equation with zero as one of the solutions? Describe the equation if there is any. You are going to think deeper and test further your understanding of solving quadratic equations by using the quadratic formula. Tell whether the solutions are real numbers or not real numbers.

Do you agree that the two equations have the same solutions? How are you going to use the quadratic formula in determining whether a quadratic equation has no real solutions?

Give at least two examples of quadratic equations with no real solutions. Find the solutions of the following quadratic equations using the quadratic formula. Make the Most Out of It! Were you able to come up with the quadratic equation that represents the area of each piece of plywood? Were you able to determine the length and width of each piece of plywood? Another quadratic equation has 2. Do you think the quadratic formula is more appropriate to use in solving quadratic equations?

Explain then give examples to support your answer. The values of a. The car covers km in three hours less than the time it takes the truck to travel the same distance.

What is the length of the side of the base of the box? The area of the car park is m2. What new insights do you have about solving quadratic equations by using the quadratic formula?

What equation represents the area of the car park? What are the dimensions of the table? How would you represent the width of the car park? How about its length?

A car travels 30 kph faster than a truck. Suppose the area of the car park is doubled. The height of the box is 4 cm. What is the length of the car park? How about its width? Grace constructed an open box with a square base out of cm 2 material.

What is the speed of the car? What about the truck? How would you use the equation representing the area of the car park in finding its length and width? If you are to solve each of the following quadratic equations.

The length of a car park is m longer than its width. The length of a rectangular table is 0. How will you do it? Draw the floor plan. Formulate as many quadratic equations using the floor plan that you prepared. You will be given a practical task which will demonstrate your understanding.

Dining room f. Solve the equations using the quadratic formula. Laundry Area 1. Luna to prepare a floor plan. He asked an architect to prepare a floor plan that shows the following: Comfort room d. Living room e. Luna would like to construct a new house with a floor area of 72 m2.

Kitchen e. Suppose you were the architect asked by Mr. The lesson provided you opportunities to describe quadratic equations and solve these by using the quadratic formula.

This lesson was about solving quadratic equations by using the quadratic formula. Which are Real? Which are Not? Refer to the numbers below to answer the questions that follow. Start lesson 3 of this module by assessing your knowledge of the different mathematics concepts previously studied and your skills in performing mathematical operations. These knowledge and skills will help you in understanding the nature of roots of quadratic equations.

How do you describe numbers that are perfect squares? Were you able to classify the given numbers as real or not real. You have done this activity in the previous lessons so I am sure you are already familiar with this. Which of the numbers are rational? Which of the numbers are perfect squares? Were you able to find the value of a. Which of the numbers are real?

Which are not real? Aside from your answer. This value will be your basis in describing the roots of a quadratic equation. Math in A. Which of the numbers above are familiar to you? Describe these numbers.

What do you think is the importance of the expression b 2 — 4ac in determining the nature of the roots of quadratic equation? You will find this out as you perform the succeeding activities. Find my Equation and Roots Directions: Using the values of a. Then find the roots of each resulting equation. Were you able to find the roots of the resulting quadratic equation? Do you think the ball can reach the height of ft. How many seconds will it take for the ball to fall to the ground?

Place Me on the Table! Answer the following.? Why not? How would you describe the roots of quadratic equation when the value of b 2 — 4ac is 0? What is the distance of the ball from the ground after 6 seconds?

Complete the table below using your answers in activities 3 and 4. Study the situation below and answer the questions that follow. How do you determine the quadratic equation having roots that are real numbers and equal? Which quadratic equation has roots that are real numbers and equal?

Irrational numbers? After how many seconds does the ball reach a distance of 50 ft. Are you ready to learn more about the nature of the roots of quadratic equations? From the activities you have done. When b2 — 4ac is equal to zero. It can be zero. Find out more about the applications of quadratic equations by performing the activities in the next section.

This value can be used to describe the nature of the roots of a quadratic equation. When b2 — 4ac is greater than zero and a perfect square. To check. If the quadratic formula is used. When b2 — 4ac is greater than zero but not a perfect square. Use these values to evaluate b2 — 4ac. When b2 — 4ac is less than zero. Use the mathematical ideas and examples presented in the preceding section to answer the activities provided.

Your goal in this section is to apply the key concepts of the discriminant of the quadratic equation. If the width of the table is p meters. Give the dimensions of the table. When do you say that the roots of a quadratic equation are real or not real numbers? How does the knowledge of the discriminant help you in determining the nature of the roots of any quadratic equation? Were you able to determine the nature of the roots of any quadratic equation?

What is My Nature? Determine the nature of the roots of the following quadratic equations using the discriminant. The length of the table has to be 1 m longer than the width. How did you determine the nature of the roots of each quadratic equation? Mang Jose wants to make a table which has an area of 6 m2. Without actually computing for the roots. Form a quadratic equation that represents the situation. Answer the following questions 1. Give examples for each. In the next section.

You are going to think deeper and test further your understanding of the nature of the roots of quadratic equations. How much of your initial ideas were found in this section?

Now that you know the important ideas about the topic. Do you agree with Danica? Justify your answer by giving at least two examples. Was it easy for you to determine the nature of the roots of the quadratic equation? Try to compare your initial ideas with the discussion in this section. Describe the roots of a quadratic equation when the discriminant is a. How do you determine the nature of the roots of quadratic equation?

How is the concept of the discriminant of a quadratic equation used in solving real-life problems? Form and describe the equation representing the situation. Will It or Will It Not? When a basketball player shoots a ball from his hand at an initial height of 2 m with an initial upward velocity of 10 meters per second. You will attach a rope to a stick and throw it over the branch. How did you come up with the equation? What will be the height of the ball after 2 seconds?

Your goal in this section is to apply your learning to real life situations. With the given conditions. You want to hang your food pack from a branch 20 ft.

You will be given tasks which will demonstrate your understanding of the discriminant of a quadratic equation. You and a friend are camping. Your friend can throw the stick upward with an initial velocity of 29 feet per second. What new insights do you have about the nature of the roots of quadratic equations?

More importantly. How did the task help you see the realworld use of the topic? The lesson provided you with opportunities to describe the nature of the roots of quadratic equations using the discriminant even without solving the equation. Then model the path of the ball by a quadratic expression. Write a similar situation but with varied initial height when the ball is thrown with an initial upward velocity.

Cite two more real-life situations where the discriminant of a quadratic equation is being applied or illustrated. Will the ball hit the ring if the ring is 3 m high? You were also asked to cite real-life situations where the discriminant of a quadratic equation is applied or illustrated. How long will it take the ball to reach the height of 4. Do you think the ball can reach the height of 12 m?

Using the situation and the quadratic expression you have written in item e. Your understanding of this lesson and other previously learned mathematical concepts and principles will facilitate your understanding of the succeeding lessons.

How long will it take to touch the ground? Perform the indicated operation then answer the questions that follow. Start lesson 4 of this module by assessing your knowledge of the different mathematics concepts and principles previously studied and your skills in performing mathematical operations. What mathematics concepts and principles did you apply to arrive at each result?

How did you determine the result of each operation? These knowledge and skills will help you in understanding the sum and product of the roots of quadratic equations.

Which quadratic equation did you find difficult to solve? Relate Me to My Roots! You may work in groups of 4. Were you able to find the roots of each quadratic equation?

Find My Roots! Find the roots of each of the following quadratic equations using any method. How did you find the roots of each quadratic equation?

Which method of solving quadratic equations did you use in finding the roots? Were you able to perform each indicated operation correctly? Do you think a quadratic equation can be determined given its roots or solutions? Justify your answer by giving 3 examples. Determine the roots of each quadratic equation using any method. A rectangular garden has an area of m2 and a perimeter of 46 m. What are the values of a. What do you observe about the sum and the product of the roots of each quadratic equation in relation to the values of a. Were you able to relate the values of a.

Complete the following table. Do you think a quadratic equation can be determined given the sum and product of its roots?

Find the roots of the equation formulated in item 1. Suppose you are asked to find the quadratic equation given the sum and product of its roots. What can you say about the equation formulated in item 1?

What is the product of the roots? How is this related to area? Were you able to relate the sum and product of the roots of a quadratic equation with its values of a.

Let us now find the sum. What equation would describe the area of the garden? Write the equation in terms of the width of the garden. What do the roots represent? What is the sum of the roots? How is this related to the perimeter? You will be able to answer this as you perform the succeeding activities. Remember that the roots of a quadratic equation can be determined using the quadratic formula.

The roots of the equation are 1 and Use these values to find the sum and the product of the roots of the equation. Then determine the sum and the product of the roots that will be obtained. Use the values of a. These values of x make the equation true. Sum of the roots: What to PROCESS Your goal in this section is to apply previously learned mathematics concepts and principles in writing and in determining the roots of quadratic equations. How did you determine the sum and the product of the roots of each quadratic equation?

Now that you learned about the sum and product of the roots of quadratic equations. Who Am I? This is My Sum and this is My Product. Verify your answers by obtaining the roots of the equation. What mathematics concepts or principles did you apply to arrive at the equation?

Are there other ways of getting the quadratic equation given the roots? If there are any. Here Are The Roots. Perform the next activity. What do you think is the significance of knowing the sum and the product of the roots of quadratic equations? Was it easy for you to determine the sum and the product of the roots of quadratic equations? Were you able to find out the importance of knowing these concepts?

Do you think it is always convenient to use the values of a. How did you determine the quadratic equation given its roots? Were you able to determine the quadratic equation given its roots? Did you use the sum and the product of the roots to determine the quadratic equation? Let us now find out how the sum and the product of roots are illustrated in real life. Where is the Trunk? How is the given situation related to the lesson. Mang Juan owns a rectangular lot.

What equation represents the perimeter of the lot? How about the equation that represents its area? What are the dimensions of the rectangular lot? Using your idea of the sum and product of roots of quadratic equation. The perimeter of the lot is 90 m and its area is m2. Fence my Lot! Read and understand the situation below to answer the questions that follow.

Method 1: You goal in this section is to take a closer look at some aspects of the topic. You are going to think deeper and test further your understanding of the sum and product of roots of quadratic equations. The following are two different ways of determining a quadratic equation whose roots are 5 and Think of These Further!

Method 2: Which method of determining the quadratic equation do you think is easier to follow? What new insights do you have about the sum and product of roots of quadratic equations? Suppose the sum of the roots of a quadratic equation is given. Write the equation. What do you think are the advantages and disadvantages of each method used in determining the quadratic equation?

Explain and give 3 examples. If one of the roots is 7. Suppose the product of the roots of a quadratic equation is given. If one of the roots is Describe each method of finding the quadratic equation.

The sum of the roots of a quadratic equation is The perimeter of a rectangular bulletin board is 20 ft. If the area of the board is 21 ft. You will be given a practical task in which you will demonstrate your understanding.

The product of the roots of a quadratic equation is Three pictures showing the applications of sum and product of roots of quadratic equations in real life.

Work in a group of 3 and make a scrap book that contains all the things you have learned in this lesson. This includes the following: In this lesson. Describe how quadratic equations are illustrated in the pictures. How did you find the performance tasks?

How did the tasks help you see the real-world use of the topic? At least 5 examples of finding the quadratic equations given the roots. A journal on how to determine a quadratic equation given the roots. Your understanding of this lesson and other previously learned mathematics concepts and principles will facilitate your learning of the succeeding lessons.

These knowledge and skills will help you in understanding the solution of equations that are transformable to quadratic equations. What method of solving quadratic equations did you use to find the roots of each? Start lesson 5 of this module by assessing your knowledge of the different mathematics concepts and principles previously studied and your skills in performing mathematical operations. These mathematical skills are necessary for you to solve equations that are transformable to quadratic equations.

Mary and Carol are doing a math project. You will learn this in the succeeding activities. How did you simplify the resulting expressions? Were you able to add or subtract the rational expressions and simplify the results? Suppose you were given a rational algebra equation. How long does it take Mary working alone to do the same project? Carol can do the work twice as fast as Mary.

How did you find the sum or the difference of the rational algebraic expressions? Read and understand the situation below. What mathematics concepts or principles did you apply in adding or subtracting the rational algebraic expressions?

Perform the indicated operation then express your answer in simplest form. But how are quadratic equations used in solving real-life problems? You will find this out in the activities in the next section. If Mary can finish the job in x hours alone. How would you describe the equation formulated in item 3? Once the equations are transformed to quadratic equations.

How would you solve the equation formulated? What mathematics concepts and principles are you going to use? Are you ready to learn more about rational algebraic equations? The different methods of solving quadratic equations.

How would you represent the amount of work that Mary can finish in 1 hour? How about the amount of work that Carol can finish in 1 hour? These equations may be given in different forms.

This is a quadratic equation that is not written in standard form. To write the quadratic equation in standard form. The given equation is a quadratic equation but it is not written in standard form. Try factoring in finding the roots of the equation. Transform this equation to standard form. Write the resulting quadratic equation in standard form. If the obtained values of x make the equation In the given equation.

Find the roots of the resulting equation using any of the methods of solving quadratic equations. The solutions of the equation are: To find the roots of the equation. Let us solve the equation by factoring. Multiply both sides of the equation by the LCM of all denominators. Check whether the obtained values of x make the equation 8 4x true.

The same is true with View Me In Another Way! Your goal in this section is to transform equations into quadratic equations and solve these. Did you find any difficulty in transforming each equation into a quadratic equation? How did you transform each equation into quadratic equation? What mathematics concepts or principles did you apply?

Which equation did you find difficult to solve? Were you able to transform each equation into a quadratic equation? Why do you think there is a need for you to do such activity? Find this out in the next activity. What mathematics concepts or principles did you apply to solve each equation? Were you able to solve the given equations? Was it easy for you to transform those equations into quadratic equations?

Do you think there are other ways of solving each equation? Show these if there are any. Find the solution set of the following.

How would you represent the amount of work that Mark can finish in 1 hour? How about the amount of work that Jessie can finish in 1 hour? Jessie and Mark are planning to paint a house together. If Mark can finish the job in m hours. Jessie thinks that if he works alone. Working together. How will you solve the equation formulated? Which of the following equations have extraneous roots or solutions? Suppose a quadratic equation is derived from a rational algebraic equation.

How do you determine the solutions of quadratic equations? How about rational algebraic equations transformable to quadratic equations? How do you check if the solutions of the quadratic equation are also the solutions of the rational algebraic equation?

How does the concept of quadratic equation used in solving real-life problems? How do you transform a rational algebraic equation into a quadratic equation?

Explain and give examples. You are going to think deeper and test further your understanding of the solution of equations that are transformable to quadratic equations. If the two pipes are opened at the same time. Use the situation to answer the following questions. In a water refilling station. What new insights do you have about this lesson?

How many minutes does each pipe take to fill the tank? What quantities are involved in the situation? Which of these quantities are known? How about the quantities that are unknown?

What do the solutions obtained represent? How is the concept of a rational algebraic equation transformable to a quadratic equation applied in the situation? Cite a real-life situation where the concept of a rational algebraic equation transformable to a quadratic equation is being applied.

You will be given tasks which will demonstrate your understanding. The situation is clear but the use of a rational algebraic equation transformable to a quadratic equation and other mathematic concepts are not properly illustrated. The situation is not so clear. The situation is not clear and the use of a rational algebraic equation transformable to a quadratic equation is not illustrated.